How do I mix two bump nodes and fade from one to the other?
Your chosen route is to mix two shaders, each shader with its own normals, generated from heights, using Bump nodes.
Mixing Shaders is, as far as I know, gathering contributions from each shader, biasing samples made by the renderer, in Cycles. In EEVEE, I'm not sure.  In particular, I don't know how normals are combined. Maybe someone who knows can help with that. Mixing normals is tricky: they represent directions; just mixing the colors that encode the directions won't do.
Why the ramble? Because, however that's done, mixing heights is easy by comparision. It's just an addition.
Personally, using the 'Node Wrangler' shipped-add-on's convenience, CtrlShift-clicking nodes to preview their output as I went along, I would ..


Preview the Wave and Noise textures, and try them out through Bump on their own, on a neutral material. I would put whatever was needed around them (perhaps Color Ramp / Multiply) to adjust the heights, before they get to  the Bump node.
Preview the Gradient, on its own, to make sure it was oriented correctly, and roughly right.

So the resulting chunk of tree would be something like this:

What's in those 'Adjusted' groups will depend on how you want your waves and noise to look. The Multiply Add  is just my choice for changing the width and location of the gradient.
The only suggestion is that it's easier to mix heights, here, than shaders, or normals.